Ta có:
`\quad S_1=u_1`
`\quad S_2=u_1+u_2`
`\qquad `…
`\quad S_{2010}=u_1+u_2+...+u_{2010}`
*`S_{2010}-S_1=u_2+u_3+...+u_{2010}`
*`S_{2010}-S_2=u_3+u_4+...+u_{2010}`
….
*`S_{2010}-S_{2009}=u_{2010}`
Ta có:
`S=2u_1+3u_2+4u_3+...+2011u_{2010}`
`S=2(u_1+u_2+...+u_{2010})+(u_2+u_3+...+u_{2010})+(u_3+u_4+...+u_{2010})+...+(u_{2009}+u_{2010})+u_{2010}`
`S=2S_{2010}+S_{2010}-S_1+S_{2010}-S_2+...+S_{2010}-S_{2008}+S_{2010}-S_{2009}`
`S=2011S_{2010}-(S_1+S_2+..+S_{2009})`
`S=2011.(3^{2010}-1)-(3^1-1+3^2-1+...+3^{2009}-1)`
`S=2011.3^{2010}-2011+1.2009-(3+3^2+...+3^{2009})`
`S=2011.3^{2010}-2-{3.(3^{2009}-1)}/{3-1}`
`S={2.2011.3^{2010}-2.2-3^{2010}+3}/2`
`S={4021.3^{2010}-1}/2`
Vậy `S={4021.3^{2010}-1}/2`